Multilevel and multifrailty models. Overview Multifrailty versus multilevel Only one cluster, two frailties in cluster e.g., prognostic index (PI) analysis,

Multilevel and multifrailty models

Overview Multifrailty versus multilevel

Only one cluster, two frailties in cluster e.g., prognostic index (PI) analysis, with random center effect and ramdom PI

effect

More than one clustering level e.g., child mortality in Ethiopia with children clustered in village and village in

district Modelling techniques

Bayesian analysis through Laplacian integration Bayesian analysis throgh MCMC Frequentist approach through numerical integration

2Overview

Bayesian analysis through Laplacian integration Consider the following model with one clustering level

with the random center effect covariate information of first covariate, e.g. PIfixed effect of first covariaterandom first covariate by cluster interaction other covariate information other fixed effects

3Bayesian analysis through Laplacian integration

Aim of Bayesian analysis: obtain posterior distributions for parameters of interest

Ducrocq and Casella (1996) proposed a method based on Laplacian integration rather than Gibbs sampling

Laplacian integration is much faster than Gibbs sampling, which makes it more suitable to their type of data, i.e., huge data sets in animal breeding, looking at survival traits

Emphasis is placed on heritability and thus estimation of variance components

Posterior distributions are only provided for variance components, fixed effects are only considered for adjustment


The joint posterior density is given by

with

the variance of the random centre effect the variance of random covariate by cluster interaction

likelihood

furthermore, we have the prior distributions

random effects are independent from each other


and for the other prior parameters flat priors are assumed

We leave baseline hazard unspecified, and therefore use partial likelihood (Sinha et al., 2003)


The logarithm of the joint posterior density is then proportional to

The posterior distribution of the parameters of interest can be obtained by integrating out other parameters


No analytical solution available for

Approximate integral for fixed value by Laplacian integration

For fixed value we use notation

First we obtain the mode of the joint posterior density function at

by maximising the logarithm of the joint posterior density wrt and (limited memory quasi-Newton method)


We first rewrite integral as

and replace by the first terms of its Taylor expansion around the mode

The second term of the Taylor expansion cancels

For the third term we need the Hessian matrix

Bayesian analysis through Laplacian integration

The integral is then approximately

As has asymptotically a multivariate normal distribution with variance covariance matrix , we have


The integral can therefore be simplified to

or on the logarithmic scale

Estimates for and are provided by the mode of this approximated bivariate posterior density

To depict the bivariate posterior density and determine other summary statistics by evaluating the integral on a grid of equidistant points


The values can be standardised by computing

with the distance between two adjacent points

Univariate posterior densities for each of the two variance components can be obtained as


This discretised version of the posterior densities can also be used to approximate the moments The first moment for is approximated by

In general, the cth moment is given by


From these non-central moments, we obtain The mean

The variance

The skewness


Bayesian analysis through Laplacian integration: Example Prognostic index heterogeneity for a bladder cancer multicentre trial Traditional method to validate prognostic index

Split database into a training dataset (typically 60% of the data) and validation dataset (remaining 40%)

Develop the prognostic index based on the training dataset and evaluate it based on the validation dataset

Flaws in the traditional method How to split the data, at random or picking complete centers? This will always work if sample sizes are sufficiently large It ignores the heterogeneity of the PI completely


We fit the following frailty model

with the overall prognostic index effect

the random hospital effect

the random hospital* PI interaction

and consider disease free survival (i.e., time to relapse or death, whatever comes first).

Parameter estimates

= 0.737 (se=0.0964)

= 0.095

= 0.016


Interpreting the parameter estimates Good prognosis in center i,

Poor prognosis in center i,

Hazard ratio in center i,

If for cluster i, , the mean of the distribution

with 95 % CI

This CI refers to the precision of the estimated conditional hazard ratio


Bivariate posterior density for


Univariate posterior densities for


Plotting predicted random center and random interactions effects


Interpretation of variance component

Heterogeneity of median event time quite meaningless, less than 50% of the patients had event, therefore rather use heterogeneity of five-year disease free percentage

Fit same model with Weibull baseline hazard

leading to the following estimate


The density of the five-year disease-free percentage usingis then given by

where and FN standard normal


The density of the five-year disease-free percentage


Interpretation of variance component The hazard ratio for center i is given by

Derive the density function for this expression.

What are the 5th and 95th quantiles for this density assuming the parameter estimates

are the population parameters


= 0.737 (se=0.0964)

= 0.016

We look for

General rule

Applied to

=lognormal with parameters and


The density function for


For lognormal distribution we have

For the x% quantile of HR, , we have

Therefore, the 5th and 95th quantiles are given by


Density function of HR is given by

The 5th and 95th quantiles for this density are thus given by 1.61 and 2.45


Documents

Multilevel and multifrailty models. Overview Multifrailty versus multilevel Only one cluster, two frailties in cluster e.g., prognostic index (PI) analysis,